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The effectiveness of Realistic Mathematics Education approach: The role of mathematical representation as mediator between mathematical belief and trouble solving

• Hutkemri Zulnaidi ,
• Effandi Zakaria

The effectiveness of Realistic Mathematics Educational activity arroyo: The role of mathematical representation as mediator between mathematical belief and problem solving

• Putri Yuanita,
• Hutkemri Zulnaidi,
• Effandi Zakaria

x

• Published: September 27, 2018
• https://doi.org/x.1371/journal.pone.0204847

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Abstruse

This study aims to identify the role of mathematical representation as a mediator between mathematical belief and trouble solving. A quasi-experimental design was developed that included 426 Course i secondary schoolhouse students. Respondents comprised 209 and 217 students in the treatment and command groups, respectively. SPSS 23.0, ANATES 4 and Amos 18 were used for data analysis. Findings indicated that mathematical representation plays a significant role as mediator betwixt mathematical conventionalities and arithmetic problem solving. The Realistic Mathematics Education (RME) approach successfully increased the arithmetics problem-solving ability of students.

Citation: Yuanita P, Zulnaidi H, Zakaria E (2018) The effectiveness of Realistic Mathematics Didactics approach: The part of mathematical representation every bit mediator betwixt mathematical belief and trouble solving. PLoS ONE 13(9): e0204847. https://doi.org/10.1371/journal.pone.0204847

Editor: Christine Due east. King, Academy of California Irvine, United states

Received: March 2, 2018; Accepted: September xiv, 2018; Published: September 27, 2018

Copyright: © 2018 Yuanita et al. This is an open admission article distributed under the terms of the Creative Eatables Attribution License, which permits unrestricted utilise, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Data files. Data availability also from author contacted at putri.yuanita@lecturer.unri.ac.id.

Funding: The authors received no specific funding for this piece of work.

Competing interests: The authors have alleged that no competing interests exist.

Introduction

Educational activity equips younger generations with important skills and knowledge. Effective learning enables students to acquire through creative teaching methods and acquire knowledge in form; the latter becomes an exciting activity through the attempt of teachers [1]. Mathematics education motivates students to become critical and innovative and to cultivate sound reasoning in problem solving. Mathematics education is an active, dynamic and continuous process; activities in mathematics education aid students develop their reasoning, recall logically, systematically, critically and thoroughly and adopt an objective and open mental attitude when dealing with problems [ii]. Teaching and learning consist of three main components, namely, teachers, students and content. Students must be equipped with cognition and loftier-level skills and teachers must possess knowledge and professionalism. Problem-solving skills enable students to recollect creatively and critically by using progressive and challenging thought processes; artistic and critical thinking will assist develop a nation and address its needs [3]. Teaching and learning processes in the classroom serve as a study basis for researchers. A future educator tin determine effective teaching methods through this procedure. Teachers and students in Indonesia admit the need to improve the current status of educational activity and learning mathematics. Since 1970, Indonesia has applied a modern approach towards pedagogy mathematics. All the same, this approach has created problematic situations in various schools.

Mathematics learning in Indonesia remains below average compared with developing countries in Asia, such as People's republic of china, Singapore and Malaysia [4]. In the by, China surpassed other western countries in internationally scaled mathematics achievement, such as in PISA and International Mathematical Olympiads (IMO) [five]. 1 of the challenges faced by mathematics pedagogy is the constantly irresolute curriculum. Traditional mathematics didactics persists in secondary schools. If the paradigm is to be changed, and so teachers must find a didactics and learning approach that is consistent with the constructivist approach. Ane of these didactics and learning approaches is Realistic Mathematics Education (RME), which was introduced in 2001 in Indonesia by the Realistic Mathematical Teaching of Indonesia (known every bit Pendidikan Matematik Realistik Indonesia or PMRI). The goal of PMRI is to revolutionise and ameliorate mathematics education [6].

The RME approach was commencement developed by the Freudenthal Institute in holland in 1971. The RME arroyo for mathematics is widely known as the best and nearly detailed approach, which was expanded from the problem-based approach for mathematics education [7]. Educational activity and learning RME take five principal criteria, namely, students' experience in daily life; changing reality to a model and changing the model through a mathematical vertical process earlier turning it into a formal organisation; use of students' active style; use of discussions and question and respond methods to cultivate the mathematics skills of students and formation of a connection between concepts and topics until learning becomes holistic and complete [8]. Since 2001, many teachers in Indonesia have been trained to use the RME arroyo. RME has been implemented in 13 of 33 provinces. On the basis of this finding, a study is conducted to develop a teaching module that uses RME and to examine the effects of teaching and learning using the mathematics learning module for secondary schools in Indonesia. Teaching and learning via RME aim to solve the problems faced by teachers and students.

The purpose of RME is to transform mathematics learning into a fun and meaningful experience for students by introducing issues within contexts. RME starts with choosing issues relevant to student experiences and knowledge [four]. The teacher then acts as a facilitator to help students solve contextual issues. This contextual problem-solving activity brings positive affect to the mathematical representation of students, which is related to their trouble solving skills [9,ten]. The all-time way to teach mathematics is to provide students with meaningful experiences by solving the problems they face every twenty-four hours or by dealing with contextual problems. Realistic mathematics educational activity enables the alteration of the mathematical fabric concept and its human relationship. Realistic mathematics pedagogy changes the culture towards a dynamic i, but still in the corridor of the educational procedure. Therefore, realistic mathematics education is an innovative learning approach that emphasises mathematics as a deed that must be associated with real life using real world context every bit the starting point of learning [11].

Mathematical belief is the fundamental thought in the application of mathematical teaching approaches [12]. The mathematical belief of a student is formed from his or her attitude towards his or her mathematical knowledge, thereby enhancing one's mathematical value. This view is supported by Anderson, Roger and Klinger [xiii], who found that positive mathematical belief influences the performance of secondary school students in Canada. According to The National College of Teachers of Mathematics (NCTM) [14], this conventionalities influences the ability of students to evaluate their skills, desire to perform mathematical tasks and mathematical disposition. Knowledge of these steps is not enough in performing mathematical tasks because students must besides believe in the truth of concepts and procedures. The mathematical belief of students consists of three chief factors, namely, students' conventionalities in their ability, in the mathematical discipline and towards mathematical educational activity and learning.

Hwang, Chen, Dung and Yang [15] defined representation every bit the process of turning a physical model in the real world into an abstract concept or symbol. In mathematical psychology, representation refers to the human relationship between objects and symbols. The five outer levels used past representation in mathematics education are real-world objects, multiple representation, arithmetic symbol representation, oral representation and picture or graphic representation. The concluding three representations are abstract and are considered high-level representation in solving arithmetics bug. Ratio with the aid of arithmetic symbol representation involves translating mathematical problems into arithmetic formulas. Language power representation involves interpreting characters and relationships in mathematical bug into verbal or vocal forms. Motion picture or graphic representation involves interpreting mathematical bug into pictures or graphics. In this study, the mathematical representations applied by students consist of motion picture representations, graphic representations, tabular representations, symbolic representations, mathematical notes, written text representations, words and language.

Trouble solving is one of the college-lodge thinking skills that require students to think critically and creatively [16]. Ibrahim [17] claimed that the ability to solve bug involves the use of learned principles to solve bug to reach certain meanings. In the nowadays study, problem solving skills refer to the ability to solve problems given in the learning context using the RME approach. The problems are based on daily routines and real situations that students were previously aware of. Trouble solving skills in this study refer to the ability of students to solve related concepts and procedures in arithmetic problems.

Problem statement

Varying education styles increases the difficulty of learning and agreement mathematics. Moreover, students are afraid of mathematics [half-dozen]. The enquiry object in mathematics is abstract and traditional education approaches are sick suited for such matters. The unsatisfactory understanding of mathematics and performances of students are attributed to several factors. Firstly, teachers dominate the learning process of a classroom by applying unidirectional and traditional didactics methods. According to Roberg [18], traditional learning focuses on skill and concept acquisition. Thus, this approach is unsuitable for improving problem solving skills. Secondly, teachers merely present theories and definitions. For instance, a theorem is explained through examples and students are assessed through a serial of exercises and questions. Didactics is the process of obtaining facts from definitions, attributes and formulas in the mathematics textbook of students. Teachers only follow the steps given in textbooks without considering whether the process is right or not. Thus, the learning process becomes mechanical, wherein teachers simply set up formulas and solutions for students [nineteen]. Findings on the application of modern mathematics show that mathematical learning is a low-value learning procedure [6].

Mathematical literacy in Indonesia cannot improve with the way mathematics is taught in schools. The current instruction approach does non focus on logical, belittling, systematic, critical and artistic thinking amidst students; rather, teachers merely depend on textbooks [20]. This arroyo requires students to memorise the correct steps for answering questions. Nevertheless, students come across difficulty when they are given questions that cannot be solved using such steps. The students learn passively and memorise formulas without understanding what the questions actually mean. Thus, they do not benefit from what they are learning and often make mistakes. Zainal [21] stated that students prefer to memorise the formulas and steps provided past their teachers without comprehending the bodily concept. Thus, students only know how to summate, only they cannot solve everyday bug that involve a mathematical concept or skill. Many students perceive that mathematics is hard to learn and requires a long time to gain understanding. Students are considered to have learned successfully when they can retrieve and recapitulate facts or use them to answer questions in examinations. Thus, students have low understanding and mastery of mathematical concepts.

According to Taat, Abdullah and Talip [22], teachers must use an arroyo that deeply influences the agreement of students. Sabandar [23] pointed out the need for challenging settings and bug to encourage students to acquire more than they used to. Mathematics is mainly trouble solving-oriented. Thus, teachers take to connect mathematics with everyday problems. To improve the problem-solving skills of students, mathematics teachers must provide open, realistic issues with multiple probable answers [24]. In realistic mathematical learning that uses open up problems, students use their problem solving methods and understand the methods used by others. This ability is important because mathematics is used in almost every attribute of life.

Few studies show the human relationship between mathematical representations and solving mathematical problems. Hwang, Chen, Dung and Yang [fifteen] mentioned that good problem-solving skills are the primal to obtaining the exact solution to a trouble. Gagatsis and Elia [25] studied the role of four-style representations, namely, verbal, decorative motion picture, informal picture and counting line representations, in solving mathematical problems. Students generally achieve meliorate problem-solving skills when the iv representation models are used than when the single-representation learning model is applied. Ling and Ghazali [26] constitute that symbols of numeric and arithmetic representations are the most often used models by students in solving issues; these symbols include answer verification from a whole set of questions. This written report must be expanded to measure samples until the findings can be generalised. Representation assessment and problem solving strategies are needed to create a specific rubric. Hwang, Chen, Dung and Yang [fifteen] studied the influence of the ability and inventiveness of various representations in mathematical problem solving using a multimedia whiteboard system. They found that the representation ability of diverse students is key to effectively solving mathematical problems. The report should be expanded from the aspect of research subjects until the findings can be generalised because the focus was not on the direct influence of representation and creativity on existent-life trouble-solving skills.

Mathematical conventionalities is i of the components of the melancholia domain, which plays a critical role in mathematical learning. The affective attribute determines student success in learning mathematics and includes mental attitude, involvement, self-concept and belief [27]. The NCTM revealed the roles of cerebral and affective aspects in mathematical learning [28]. Both aspects are influential in the mathematical operation of students. Student belief in mathematics can influence the view towards mathematical discipline, which is related to mathematical instruction and learning [3]. According to Kloosterman [29], many students take strong mathematical belief. Mathematical conventionalities attracted the attention of many educational mathematics researchers, peculiarly in other countries. Nevertheless, merely a few studies were conducted in Indonesia on the mathematical conventionalities of students. The mathematical conventionalities of students tin can be improved through the teaching method practical by teachers. Lee, Zeleke and Mavrotheris [xxx] studied the development of student belief, which tin be expanded to the influence of the students' condition and setting. Greer, Verschaffel and de Corte [31] believed that the mathematical belief of students is influenced by teachers, textbooks, learning strategy and the use of bug that exist in their environs during learning activities. Interrelated factors influence changes in students' mathematical conventionalities. Therefore, all related factors should be considered to increase the mathematical belief of students.

Arithmetic is 1 of the mathematical learning topics practical in daily life. Students experience difficulty in agreement arithmetics-related problems. The concept acquired by students is not formed by the students solely. Hence, students neglect to retain the concept in their memory. Once students learn a new concept, they forget the one-time ane. Many students do non solve problems by understanding the concept and rely instead on intuition or memorisation. Many everyday issues can be solved using comparing to facilitate the selection of contextual issues as a first step of the learning process. This step enables students to form their concepts, principles and mathematical procedures related to the topic. In accord with the objective of mathematical learning, which is to prepare students to use mathematics and its style of thinking in daily life, nosotros try to develop an arithmetic module that fits the RME arroyo. According to Sunismi [32], the learning approach and increased cognitive development showed the presence of interaction in the understanding of mathematical concepts in solving problems for Form 2 secondary school students. Haji [33] mentioned the lack of significant interaction between the approach and ability of students to solve problems.

Other studies revealed the RME function in mathematics learning. The written report unveiled the relationship amidst mathematical representation and belief and trouble-solving skills. Warsito, Darhim and Herman [9] examined the effect of RME on improving mathematical representation power. Meika, Suryadi and Darhim [x] practical RME in students' errors in solving combinatoric issues. Yuanita and Zakaria [34] investigated the differences in the mathematical belief of students based on their abilities in RME and students enrolled in regular classes. The results of the previous written report showed that RME can be effectively used to predict the mathematical representation, belief and problem-solving skills of students. A previous study suggested a highly constructive learning approach in RME; this approach includes designing instructional materials in accordance with real-life contexts that railroad train student thinking skills. Mathematical learning should be delivered in a form that gives students an opportunity to reinvent ideas and mathematical concepts along with teacher guidance through exploration of various contextual issues and the furnishings of RME on students' mental attitude, problem-solving ability, learning interest or other variables related to mathematics learning.

Radzali, Meerah and Zakaria [35] examined the human relationship between mathematical belief and representation with mathematical problem solving. Results testify that mathematical belief and representation contributed to the problem solving of students. The findings of this study are important because no other study has examined the factors mentioned. A previous study focused on examining each separately stated gene. However, studies that contain all three factors into within or outside of the country are defective. Therefore, the electric current study attempts to investigate these three factors simultaneously to identify the effect of mathematical representation as a mediator between mathematical belief and trouble solving.

The significance of this study is its accent on mathematical representation, mathematical belief and problem-solving skills, which are vital to building mathematical discipline. Mathematical representation and belief and problem-solving skills are oftentimes misconceived. Therefore, the use of RME in the classroom can provide examples for students based on their daily activities. This approach could assist them in mathematical representation and belief and improve their problem-solving skills. Thus, this written report investigates the difference in mathematical representation and conventionalities and problem-solving skills of students who learned with RME and students who were engaged in conventional learning. This study also investigated the outcome of mathematical representation as a mediator between mathematical belief and problem solving.

Fig 1 shows that this study was performed to identify the effectiveness of the RME approach in mathematical conventionalities and representation and problem solving. In add-on, this study identified the role of mathematical representation as a mediator betwixt mathematical belief and problem solving. This report was conducted to respond the following inquiry questions:

1. Does the utilize of the RME approach have any pregnant effect on mathematical conventionalities, mathematical representation and problem solving?
2. Is mathematical representation a significant mediator between mathematical belief and trouble solving?

Methodology

Participants

The report involved 426 Course 1 secondary school students, who were divided into control and treatment groups. RME and traditional approaches were used past 209 and 217 students, respectively. The treatment grouping had 95 male and 114 female students. Fifty-six students had low ability, 96 had boilerplate ability and 57 students had high ability. The command group had 103 male and 114 female students. Sixty of them had low power, 96 had boilerplate power and 61 students had high ability. The mathematics ability of students was based on the results of their mathematics achievement in the past semester. The results were and then categorised using Anates software into low, moderate and loftier [36]. The demographic profile is shown in Tabular array 1.

Research pattern

The study used the quasi-experimental design with non-equivalent pre- and post-exam command groups. The control grouping was created for comparing with the experimental group [37,38]. The quasi-experimental design refers to an experiment that consisted of units with handling. This approach was utilised because the study used the existing class [39], which indicated that the research subjects were not selected randomly [twoscore]. The quasi-experimental pattern was used to determine the effectiveness of the RME approach in improving problem solving skills, mathematical representation and belief of students. The inquiry pattern is shown in Table 2.

Pre- and post-tests were conducted in both groups. The pre-test ensured similarity betwixt groups and statistical control by comparison the mean of mathematical belief, representation, and problem solving with significant value of more than 0.05. The treatment grouping was given a chore using the RME approach in teaching, whereas the traditional method was used equally control group. Students in both groups were taught during 10 two-hour sessions in their respective classrooms. The post-test was given to both groups afterwards they were taught social arithmetic to make up one's mind the effectiveness of the RME approach. The examination questions for pre- and post-tests were like. The researcher observed each session for both groups throughout the discussion. Observations were conducted for five weeks in 10 sessions for both groups. A post-exam was given to the two groups after social arithmetic and ratio were taught.

Internal and external validities were determined with reference to Johnso and Christensen [40]. Internal validity is a controlled variable fix by the researcher that aims to identify the actual effect on the treatment variable. External validity sees how far the findings tin can be applied to individuals and settings other than the ones in the study. Bug, such equally pick of research and lost subjects (bloodshed), emotional maturity, intellectual and physical well-being, testing, research musical instrument and validity of research objects, can arise from the quasi-experimental design of pre- and post-tests. These issues refer to factors related to the report and the attitude and emotion of students.

Experimental grouping

The experimental group was taught using the RME approach. Teachers followed 3 chief phases to teach this approach. In the first phase, teachers introduced realistic problems to students and helped them empathize the problem setting. Teachers revised previous concepts and continued them with the experience of students. In the 2d phase, students worked in groups. Each student had a book that independent contextual questions and synthetic situational problems, shared ideas, analysed patterns, made guesses and expanded problem-solving strategies based on cognition or formal feel. The tertiary phase of assessment showed the progress of students in trouble solving. They discussed their problems and discovered useful strategies. Teachers guided and instructed students throughout the discussion on how to solve issues efficiently and effectively.

Traditional group

Students in the command grouping were taught using a marking and whiteboard. They participated in the exercises given past the teachers. The exercises are based on reference books provided by the school. Each school uses dissimilar reference books. Teachers narrated and jotted downward information on the whiteboard. The enhanced educational curriculum unit of measurement requires every teaching method to exist contextual. Thus, all teachings conducted in low secondary schools are traditionally contextual didactics.

Preparation for teachers

Vi teachers were involved in the RME approach. They were selected based on the criteria of the RME approach training organised by the Ministry building of Pedagogy in Indonesia. The teachers underwent training for one month to ensure the success of the report and consistency with the pattern program. The written report objectives, RME and traditional approaches, planning and execution process and assessment methods were introduced to the teachers. The same teachers were assigned to treatment and command groups. The report was conducted after they understood the entire concept. The researcher observed throughout the written report to determine whether the teachers were using the RME approach. Observation began from the first until the end of class for every session. The teachers were given feedback well-nigh their teaching. The researcher observed the traditional class to ensure that the teachers were not using the RME approach or any other teaching method.

Pilot study

The nowadays study was reviewed and canonical by the Ministry building of Instruction Pekanbaru City, Riau, Indonesia. A airplane pilot study was conducted with 100 students to determine the validity and reliability of the research instrument. The validity of the research instrument was verified past four experts; two experts for content and 2 for language. Co-ordinate to the experts, the instrument language is suitable for measuring mathematical belief, representation and problem solving. The data from the pilot report were analysed using SPSS 23.0 and ANATES 4. Findings showed that the reliability of the mathematical conventionalities instrument, problem solving and mathematical representation are 0.93, 0.87 and 0.80, respectively. The discriminant and difficulty alphabetize for the mathematical belief examination and the mathematical problem solving test are at good and an average levels, respectively. [36] stated that the difficulty index value is at its best when used at the average level. The discriminant index should be at good and very good levels. The pilot report results indicated that the developed items are solid and strong for the bodily study.

Measure

Mathematical conventionalities instrument.

The instrument of mathematical belief was adjusted from the Mathematical Problem Solving Beliefs Instrument [41] and students' mathematics-related beliefs questionnaire [42]. The latter measures three factors of students' mathematical belief, which are related to students in terms of mathematics students, mathematical discipline, mathematical teaching and learning. Sixty statements in the mathematical belief calibration were used. Each statement could exist answered with 5 responses of strongly agree (SA), agree (A), slightly disagree (SD), disagree (D) and strongly disagree (SLD).

Mathematical representation instrument.

The instruments for mathematical representation consisted of a written test ready with iv questions on the topic of arithmetics. The instrument was constructed past the researcher to collect information about a representation problem solved by the students and their success in solving mathematical bug. This instrument had four problem statements with an open-question format. These mathematical problems required students to utilise comprehension, analysis and interpretation in the context of daily life. The total score for each item was 4 and 0 was the everyman score.

Problem solving instrument.

The Mathematical Problem Solving Beliefs Musical instrument is used to collect information about the method and the success of how the students solve mathematical problems. This musical instrument has five problem statements with an open-question format and requires students to comprehend, analyse and translate these problems in the context of daily life. The full score for each item is four and 0 is the lowest score. The problem solving instrument is measured using marker schemes. The full score for each item is 4 and 0 is the lowest score. The total score of the students is inverse to a scale of 0 to 100. The mark scheme for each item is shown in Tabular array 3.

The marking scheme used for levels of mathematical representation and trouble solving is the same every bit that used by [43], which was adapted to the arrangement outlined past the regime.

Data analysis

The analysis for the bodily written report was performed using SPSS 23.0 and Amos 18. Assay of covariance (ANCOVA) was performed to place the deviation in mathematical conventionalities, representation and problem solving betwixt the treatment and the control groups where the pre-test is a covariate. This step was followed in the structural equation modelling (SEM) test to place the role of mathematical representation as a significant mediator in the relationship between mathematical belief and problem solving.

Research findings

Difference in mathematical conventionalities proceeds score betwixt treatment and control groups

Univariate Analysis of Variance (UNIANOVA) was performed to identify the gain scores of the mathematical belief of the treatment and the command groups. Certain requirements for the exam needed to be met prior to UNIANOVA. These requirements include normality and homogeneity of variance between groups. The normality test showed the skewness and kurtosis values for the mathematical belief gain score for the handling and the control groups are (0.07, -0.82) and (-0.36, 0.32), respectively. This outcome shows that normality requirement was met and data were considered normal if the skewness and kurtosis value ranged from -i.96 to +1.96 [44]. Therefore, one-mode UNIANOVA can be performed to identify the differences in the mathematical belief gain score of the handling and the control groups, as shown in Table 4.

The UNIANOVA test result in Table 4 shows a significant deviation in the mathematical belief gain score betwixt the treatment and the control groups [F = 39.963, sig = 0.000 (p < 0.05)]. Students in the handling grouping (mean = 0.606, std. mistake = 0.07) have a higher mathematical conventionalities than students in the control group (hateful = -0.027, std. error = 0.07). This finding means that the RME approach has better result on the increase in the mathematical belief of students than the use of the traditional method. This differential effect size is medium (Cohen's d = 0.61) [45]. Inspection of the 95% confidence intervals around each mean indicated that a significant increase in mathematical conventionalities for participants in the treatment grouping and no increase in mathematical conventionalities for participants in the control group, equally shown in Table 5.

Fig 2 shows the pre- and postal service-exam ways for a 2-grouping design. In the handling group, mail service-test results (mean = 3.90) had higher mathematical conventionalities than pre-examination results (mean = 3.29). Nonetheless, in the control grouping, pre-test results (hateful = three.23) had college mathematical conventionalities than postal service-test results (hateful = 3.21).

Deviation in mathematical representation gain score of handling and control groups

UNIANCOVA was performed to place the deviation between the mathematical representation gain score of the treatment and the control groups. The normality test showed the skewness and kurtosis values for mathematical representation pre-exam for the handling (0.09, -0.57) and the control (-0.05, -0.78) groups. These results indicated that the normality requirement was met. Levene'south test obtained F = 1.525, sig = 0.434 (p > 0.05), which showed that the information had similar variances between groups. Thus, UNIANCOVA tin can exist performed to identify the difference in mathematical representation proceeds scores between the treatment and the command groups.

The UNIANCOVA exam outcome in Table 6 showed no significant departure between the mathematical representation gain score of the treatment and the control groups [F = 0.430, sig = 0.512 (p > 0.05)]. The mathematical representation gain score of the students in the handling group (hateful = 1.17) was similar to that of the students in the control group (mean = i.23). This event indicated that the RME arroyo and the traditional method had the same effect on the increment in the mathematical representation of students. This differential upshot size was small (Cohen'southward d = 0.06) [45].

Fig iii shows the pre- and post-test means for a two-group design. In the treatment group, mail service-test results (mean = 2.90) had higher mathematical representation than pre-test results (mean = one.73). However, in the control group, post-exam results (hateful = 2.74) had higher mathematical representation than pre-examination results (hateful = 1.52).

Differences in mathematical problem-solving gain scores of treatment and control groups

UNIANCOVA was performed to identify the difference betwixt mathematical problem-solving gain scores of the treatment and the control groups. The normality test showed the skewness and kurtosis values of mathematical problem-solving gain scores for the treatment group (-0.27. -0.81) and the command grouping (0.38, -0.48). Results showed that the normality requirement was met. Levene's exam obtained a value of F = 1.440, sig = 0.231 (p > 0.05), which indicated that the data had similar variances between groups. Therefore, UNIANCOVA can be performed to place the differences betwixt mathematical problem-solving gain scores of the treatment and the control groups, every bit shown in Table 7.

The UNIANCOVA exam consequence in Table 7 showed a significant difference in mathematical trouble-solving gain scores betwixt the handling and the command groups [F = 6.716, sig = 0.010 (p < 0.05)]. Students in the treatment grouping (mean = two.01) had better mathematical trouble solving gain scores than the students in the command group (mean = 1.85). These results prove that the RME approach was better than the traditional method at improving problem solving skills. Such differential effect size was small-scale (Cohen's d = 0.25) [45].

Fig 4 shows the pre- and post-examination means for a two-group design. In the treatment group, postal service-test results (mean = 2.lxx) had higher mathematical problem-solving value than pre-examination results (mean = 0.68). Nonetheless, in the control group, post-examination results (mean = 2.39) had college mathematical problem-solving values than pre-test results (mean = 0.54).

Role of mathematical representation as a mediator between mathematical belief and problem solving for the treatment group

SEM analysis was performed to identify the role of arithmetics representation as a mediator between belief towards mathematical education and learning and mathematical problem solving. The analysis result of the SEM path model in Fig 5 shows the following: chi square/df = iii.06, root hateful-square error approximation (RMSEA) = 0.07, goodness of fit index (GFI) = 0.91, Tucker–Lewis fit alphabetize (TLI) = 0.90 and comparative fit index (CFI) = 0.92. All assessments indicated that the information in the study had reasonable aligning for the suggested model [46]. The consequence of SEM analysis showed that the suggested regression model was suitable when mathematical teaching conventionalities (β = 0.33, p < 0.05) and mathematical learning belief (β = 0.52, p < 0.05) are significant predictor variables for mathematical problem solving. The SEM result showed that mathematical teaching belief (β = 0.52, p < 0.05) and mathematical learning belief (β = 0.70, p < 0.05) are significant predictor variables for arithmetic representation. Bootstrapping examination was performed to determine the effect of mathematical representation every bit a significant mediator.

Bootstrapping examination was applied to decide the effect of arithmetic representation equally a significant mediator betwixt mathematical teaching and learning conventionalities and problem solving. Table viii shows that arithmetics representation is a significant fractional mediator between educational activity belief (β = 0.19, p < 0.05) and learning (β = 0.29, p < 0.001) towards trouble solving.

Role of mathematical representation every bit a mediator between mathematical belief and problem solving for the control group

SEM analysis was performed to identify the role of arithmetic representation as a mediator between the belief towards mathematical didactics and learning in mathematical problem solving. The analysis of the SEM path model in Fig 6 shows the mensurate of chi foursquare/df = 1.31, RMSEA = 0.07, GFI = 0.91, TLI = 0.xc and CFI = 0.92. The result of SEM analysis indicated that the suggested regression model was suitable when mathematical teaching belief (β = 0.36, p < 0.05) and mathematical learning conventionalities (β = 0.57, p < 0.05) were meaning predictor variables for mathematical problem solving. The SEM outcome showed that mathematical teaching conventionalities (β = 0.57, p < 0.05) and mathematical learning conventionalities (β = 0.74, p < 0.05) were significant predictor variables for arithmetic representation. Bootstrapping test was conducted to determine the effects of mathematical representation as a significant mediator (Tabular array 9).

The bootstrapping test was applied to check the effect of arithmetic representation as a significant mediator betwixt mathematical teaching and learning belief and problem solving. Table nine shows that arithmetic representation was a significant mediator for teaching belief (β = 0.19, p < 0.001) and learning (β = 0.25, p < 0.001) towards trouble solving. The SEM result indicated that the treatment and the control groups obtained the same results for the role of mathematical representation equally a fractional mediator between mathematical conventionalities and trouble solving.

Word

Students who were taught using the RME approach had higher mathematical belief than students who were exposed to the traditional method. The use of RME increased the confidence of students in mathematics, especially in arithmetics, as reflected in their active participation in the activities presented with the RME approach. According to Fauzan [47], active students employ the RME approach, which develops artistic thinking and lessens uncertainty towards mathematics. Withal, the use of the traditional method successfully increased the mathematical belief of students, although the RME approach had better upshot. Saragih [48] stated that the reward of the RME approach is its power to strengthen students' interest in mathematics. The findings supported Lee, Zeleke and Mavrotheris [30] who asserted that the RME approach enables students to larn mathematics actively such that their belief can increase through the endeavour of teachers. Greer, Verschaffel and de Corte [31] supported this idea past stating that the mathematical conventionalities of students is influenced by factors, such as teachers, textbooks, learning strategies and use of issues that exist in the surroundings of students for learning activities.

The use of the RME approach did not significantly increase mathematical representation compared with the traditional method. Thus, the RME approach was non suitable for all skills or topics. However, the RME approach even so successfully increased the mathematical representation of students. This idea was supported by Arsaythamby and Zubainur [49] who claimed that not all learning activities of students should be conducted using the RME arroyo. Teaching with the RME approach provided students with the opportunities to come up up with ideas that tin can enable them to solve mathematical problems easily. The traditional method provided opportunities for students to generate ideas, but these opportunities are fewer than those offered by the RME approach. Neria and Amit [l] mentioned that questions on mathematical representation are given to students to allow them to present situational bug in the form of mathematical notes, numerals, symbols, graphics, tables and pictures, which they will try to solve later. Therefore, the skills of teachers in using the RME approach must increase the mathematical representation of students to guide their gradual learning according to levels.

The RME arroyo successfully improved the problem-solving skills of students and was better than the traditional method in this aspect. In the RME arroyo, teachers checked the answers of students past writing down detailed answers and providing reasons or explanations as to how the reply was obtained. Moreover, students were motivated to stand in forepart of the form and explain their work. Jones, Thornton and Nisbet [51] found that the RME approach is suitable for arithmetic learning until the students go more confident in solving bug. This statement supported the findings of Viholainen, Asikainen and Hirvonen [52], who stated that confidence in mathematics has strong influence on mathematical problem solving and determines how a student chooses the arroyo, technique and strategy to utilise. The results of study supported Laurens, Batlolona, Batlolona and Leasa [iv], who claimed students who were taught with RME accomplished ameliorate results than the students who were involved in conventional learning.

The SEM test showed the aforementioned match betwixt the handling and the control groups, wherein mathematical representation was a significant fractional mediator betwixt mathematical conventionalities and trouble solving. Findings showed that mathematical belief indirectly affected mathematical problem-solving skills. This written report indicated no significant difference in mathematical representation, merely the mediator outcome of mathematical representation between treatment and control groups was the same. This effect suggests that mathematical representation is an indirectly important aspect in students to enhance the relationship betwixt mathematical beliefs and problem solving. The use of different methods did non influence the effect of mathematical representation as the mediator of the human relationship between mathematical behavior and problem solving. The findings supported Hwang, Chen, Dung and Yang [15] in their claim that mathematical representation contributes to the ability of students to solve mathematical problems. This study supported Ling and Ghazali [26], who reported that arithmetics is the most frequently used representation model by students in problem solving, including answer verification from all the given questions. Moreover, mathematical conventionalities affects the mathematical representation and problem-solving skills of students. This finding ways that if students believe in mathematical teaching and learning, then they will possess mathematical representation and reliable problem-solving skills. This statement is consistent with the findings of [3], who found that the belief of students towards mathematics tin influence their view on mathematical field of study, which is related to mathematical teaching and learning. The SEM results showed higher connection of mathematical conventionalities and mathematical representation in problem solving with the use of the RME approach than with the use of the traditional method. This finding is supported in Muchlis [53] and in Husna and Saragih [54].

The study successfully proved that the RME approach had a positive effect on mathematical conventionalities, representation and trouble solving amongst students. Thus, teachers need to suit their teaching methods using RME and encourage students to participate in activities and engage in discussions. The RME approach provides students with the opportunity to generate cognition on the topics that they have been taught. Students tin convey their ideas until they can form concepts for each learning step. Many students provide solutions that consist of different steps but accept the same answer. Students believe in producing results that they obtain by themselves, which is a procedure that they will later find as an arithmetic concept. School administrators must help teachers in eliminating the negative perception towards instruction and learning mathematics. The effectiveness of RME offers an opportunity to utilize the approach continuously to teach other topics for secondary schoolhouse students equally a whole. Future studies can examine the utilise of RME at various educational levels to obtain detailed information.

The contribution of this report is the identification of various learning methods oft used by students in everyday life that can be utilised to ameliorate the quality in learning through the creativity of teachers. In boosted, the RME approach is among the near effective approaches in fostering mathematical representation, conventionalities and trouble-solving skills that could improve educatee accomplishment. Few studies examined the relationship of mathematical representation as a mediator between mathematical belief and trouble solving. The present written report filled the gap by producing a new grade of relationship model through a quasi-experimental design.

The findings and results of this report provided information on the differences in mathematical representation, belief and problem-solving skills of students who learned through RME and conventional learning methods. Mathematics teachers should utilize RME in the classroom to make abstruse mathematical concepts more understandable. Teachers should be creative and innovative in designing learning with this approach. Teachers should develop boosted learning media, strategies or models that are more than suitable with learning materials or with the contexts of students. Further, schools should create contextual environments that are rich in data on means to solve real life bug.

Conclusion

The use of RME can increment mathematical belief, representation and problem solving skills. This approach successfully trains students to formulate their own ideas from real-life situations or experiences. Teachers must be encouraged to use the RME approach in pedagogy and learning mathematics. Efforts pertaining to mathematical representation should exist doubled to increase the mathematical problem solving skills of students. The belief of students is another major gene in increasing mathematical problem solving skills. Cooperation from all sides should be improved to encourage the utilize of the RME approach in teaching and learning mathematics at all schoolhouse levels to increase mathematical belief, representation and problem solving. This study seeks to serve equally a stepping stone for hereafter studies to expand the apply of the RME approach from the national to the international level.

Supporting information

Acknowledgments

The authors wish to thank Mrs. Dewi Marianti, Mrs. Suarni, Mrs. Arnidar, Mrs. Nurwahyu, Mrs. Furqonati, Mrs. Gusniwati and Mrs. Yulmaliza for their assistance with data collection.

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Source: https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0204847

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